"""simple docstring"""


from typing import Optional, Tuple, Union

import torch
from einops import rearrange, reduce

from diffusers import DDIMScheduler, DDPMScheduler, DiffusionPipeline, ImagePipelineOutput, UNetaDConditionModel
from diffusers.schedulers.scheduling_ddim import DDIMSchedulerOutput
from diffusers.schedulers.scheduling_ddpm import DDPMSchedulerOutput


a				=8





def      lowerCamelCase_					(   __lowerCAmelCase  ,       __lowerCAmelCase=BITS							)				->   List[Any]:




						'''simple docstring'''



						lowerCamelCase__   =x.device

						lowerCamelCase__   =(x * 255).int().clamp(0  ,       255							)

						lowerCamelCase__   =2 ** torch.arange(bits - 1  ,       -1  ,       -1  ,       device=__lowerCAmelCase							)
						lowerCamelCase__   =rearrange(__lowerCAmelCase  ,       "d -> d 1 1"							)
						lowerCamelCase__   =rearrange(__lowerCAmelCase  ,       "b c h w -> b c 1 h w"							)

						lowerCamelCase__   =((x & mask) != 0).float()
						lowerCamelCase__   =rearrange(__lowerCAmelCase  ,       "b c d h w -> b (c d) h w"							)
						lowerCamelCase__   =bits * 2 - 1
						return bits





def      lowerCamelCase_					(   __lowerCAmelCase  ,       __lowerCAmelCase=BITS							)				->   List[Any]:




						'''simple docstring'''



						lowerCamelCase__   =x.device

						lowerCamelCase__   =(x > 0).int()
						lowerCamelCase__   =2 ** torch.arange(bits - 1  ,       -1  ,       -1  ,       device=__lowerCAmelCase  ,       dtype=torch.intaa							)

						lowerCamelCase__   =rearrange(__lowerCAmelCase  ,       "d -> d 1 1"							)
						lowerCamelCase__   =rearrange(__lowerCAmelCase  ,       "b (c d) h w -> b c d h w"  ,       d=8							)
						lowerCamelCase__   =reduce(x * mask  ,       "b c d h w -> b c h w"  ,       "sum"							)
						return (dec / 255).clamp(0.0  ,       1.0							)





def      lowerCamelCase_					(   self  ,       __lowerCAmelCase  ,       __lowerCAmelCase  ,       __lowerCAmelCase  ,       __lowerCAmelCase = 0.0  ,       __lowerCAmelCase = True  ,       __lowerCAmelCase=None  ,       __lowerCAmelCase = True  ,       )				->   Union[DDIMSchedulerOutput, Tuple]:




						'''simple docstring'''



						if self.num_inference_steps is None:
												raise ValueError(
												    "Number of inference steps is 'None', you need to run 'set_timesteps' after creating the scheduler"							)

						# See formulas (12) and (16) of DDIM paper https://arxiv.org/pdf/2010.02502.pdf
						# Ideally, read DDIM paper in-detail understanding

						# Notation (<variable name> -> <name in paper>
						# - pred_noise_t -> e_theta(x_t, t)
						# - pred_original_sample -> f_theta(x_t, t) or x_0
						# - std_dev_t -> sigma_t
						# - eta -> η
						# - pred_sample_direction -> "direction pointing to x_t"
						# - pred_prev_sample -> "x_t-1"

						# 1. get previous step value (=t-1)
						lowerCamelCase__   =timestep - self.config.num_train_timesteps // self.num_inference_steps

						# 2. compute alphas, betas
						lowerCamelCase__   =self.alphas_cumprod[timestep]
						lowerCamelCase__   =self.alphas_cumprod[prev_timestep] if prev_timestep >= 0 else self.final_alpha_cumprod

						lowerCamelCase__   =1 - alpha_prod_t

						# 3. compute predicted original sample from predicted noise also called
						# "predicted x_0" of formula (12) from https://arxiv.org/pdf/2010.02502.pdf
						lowerCamelCase__   =(sample - beta_prod_t ** 0.5 * model_output) / alpha_prod_t ** 0.5

						# 4. Clip "predicted x_0"
						lowerCamelCase__   =self.bit_scale
						if self.config.clip_sample:
												lowerCamelCase__   =torch.clamp(__lowerCAmelCase  ,       -scale  ,       __lowerCAmelCase							)

						# 5. compute variance: "sigma_t(η)" -> see formula (16)
						# σ_t = sqrt((1 − α_t−1)/(1 − α_t)) * sqrt(1 − α_t/α_t−1)
						lowerCamelCase__   =self._get_variance(__lowerCAmelCase  ,       __lowerCAmelCase							)
						lowerCamelCase__   =eta * variance ** 0.5

						if use_clipped_model_output:
												# the model_output is always re-derived from the clipped x_0 in Glide
												lowerCamelCase__   =(sample - alpha_prod_t ** 0.5 * pred_original_sample) / beta_prod_t ** 0.5

						# 6. compute "direction pointing to x_t" of formula (12) from https://arxiv.org/pdf/2010.02502.pdf
						lowerCamelCase__   =(1 - alpha_prod_t_prev - std_dev_t**2) ** 0.5 * model_output

						# 7. compute x_t without "random noise" of formula (12) from https://arxiv.org/pdf/2010.02502.pdf
						lowerCamelCase__   =alpha_prod_t_prev ** 0.5 * pred_original_sample + pred_sample_direction

						if eta > 0:
												# randn_like does not support generator https://github.com/pytorch/pytorch/issues/27072
												lowerCamelCase__   =model_output.device if torch.is_tensor(__lowerCAmelCase							) else "cpu"
												lowerCamelCase__   =torch.randn(model_output.shape  ,       dtype=model_output.dtype  ,       generator=__lowerCAmelCase							).to(__lowerCAmelCase							)
												lowerCamelCase__   =self._get_variance(__lowerCAmelCase  ,       __lowerCAmelCase							) ** 0.5 * eta * noise

												lowerCamelCase__   =prev_sample + variance

						if not return_dict:
												return (prev_sample,)

						return DDIMSchedulerOutput(prev_sample=__lowerCAmelCase  ,       pred_original_sample=__lowerCAmelCase							)





def      lowerCamelCase_					(   self  ,       __lowerCAmelCase  ,       __lowerCAmelCase  ,       __lowerCAmelCase  ,       __lowerCAmelCase="epsilon"  ,       __lowerCAmelCase=None  ,       __lowerCAmelCase = True  ,       )				->   Union[DDPMSchedulerOutput, Tuple]:




						'''simple docstring'''



						lowerCamelCase__   =timestep

						if model_output.shape[1] == sample.shape[1] * 2 and self.variance_type in ["learned", "learned_range"]:
												lowerCamelCase__   , lowerCamelCase__   =torch.split(__lowerCAmelCase  ,       sample.shape[1]  ,       dim=1							)
						else:
												lowerCamelCase__   =None

						# 1. compute alphas, betas
						lowerCamelCase__   =self.alphas_cumprod[t]
						lowerCamelCase__   =self.alphas_cumprod[t - 1] if t > 0 else self.one
						lowerCamelCase__   =1 - alpha_prod_t
						lowerCamelCase__   =1 - alpha_prod_t_prev

						# 2. compute predicted original sample from predicted noise also called
						# "predicted x_0" of formula (15) from https://arxiv.org/pdf/2006.11239.pdf
						if prediction_type == "epsilon":
												lowerCamelCase__   =(sample - beta_prod_t ** 0.5 * model_output) / alpha_prod_t ** 0.5
						elif prediction_type == "sample":
												lowerCamelCase__   =model_output
						else:
												raise ValueError(F'''Unsupported prediction_type {prediction_type}.'''							)

						# 3. Clip "predicted x_0"
						lowerCamelCase__   =self.bit_scale
						if self.config.clip_sample:
												lowerCamelCase__   =torch.clamp(__lowerCAmelCase  ,       -scale  ,       __lowerCAmelCase							)

						# 4. Compute coefficients for pred_original_sample x_0 and current sample x_t
						# See formula (7) from https://arxiv.org/pdf/2006.11239.pdf
						lowerCamelCase__   =(alpha_prod_t_prev ** 0.5 * self.betas[t]) / beta_prod_t
						lowerCamelCase__   =self.alphas[t] ** 0.5 * beta_prod_t_prev / beta_prod_t

						# 5. Compute predicted previous sample µ_t
						# See formula (7) from https://arxiv.org/pdf/2006.11239.pdf
						lowerCamelCase__   =pred_original_sample_coeff * pred_original_sample + current_sample_coeff * sample

						# 6. Add noise
						lowerCamelCase__   =0
						if t > 0:
												lowerCamelCase__   =torch.randn(
												    model_output.size()  ,       dtype=model_output.dtype  ,       layout=model_output.layout  ,       generator=__lowerCAmelCase							).to(model_output.device							)
												lowerCamelCase__   =(self._get_variance(__lowerCAmelCase  ,       predicted_variance=__lowerCAmelCase							) ** 0.5) * noise

						lowerCamelCase__   =pred_prev_sample + variance

						if not return_dict:
												return (pred_prev_sample,)

						return DDPMSchedulerOutput(prev_sample=__lowerCAmelCase  ,       pred_original_sample=__lowerCAmelCase							)






class   __UpperCAmelCase		(      __lowerCAmelCase      ):

				def __init__( self      ,					_lowerCamelCase      ,					_lowerCamelCase      ,					_lowerCamelCase = 1.0      ,					):
										super().__init__()
										lowerCamelCase__   =bit_scale
										lowerCamelCase__   =(
										    ddim_bit_scheduler_step if isinstance(_lowerCamelCase      ,					_lowerCamelCase ) else ddpm_bit_scheduler_step
										)

										self.register_modules(unet=_lowerCamelCase      ,					scheduler=_lowerCamelCase )






				@torch.no_grad()
				def __call__( self      ,					_lowerCamelCase = 256      ,					_lowerCamelCase = 256      ,					_lowerCamelCase = 50      ,					_lowerCamelCase = None      ,					_lowerCamelCase = 1      ,					_lowerCamelCase = "pil"      ,					_lowerCamelCase = True      ,					**_lowerCamelCase      ,					):
										lowerCamelCase__   =torch.randn(
										    (batch_size, self.unet.config.in_channels, height, width)      ,					generator=_lowerCamelCase      ,					)
										lowerCamelCase__   =decimal_to_bits(_lowerCamelCase ) * self.bit_scale
										lowerCamelCase__   =latents.to(self.device )

										self.scheduler.set_timesteps(_lowerCamelCase )

										for t in self.progress_bar(self.scheduler.timesteps ):
																# predict the noise residual
																lowerCamelCase__   =self.unet(_lowerCamelCase      ,					_lowerCamelCase ).sample

																# compute the previous noisy sample x_t -> x_t-1
																lowerCamelCase__   =self.scheduler.step(_lowerCamelCase      ,					_lowerCamelCase      ,					_lowerCamelCase ).prev_sample

										lowerCamelCase__   =bits_to_decimal(_lowerCamelCase )

										if output_type == "pil":
																lowerCamelCase__   =self.numpy_to_pil(_lowerCamelCase )

										if not return_dict:
																return (image,)

										return ImagePipelineOutput(images=_lowerCamelCase )



"""simple docstring"""


from math import sqrt





def      lowerCamelCase_					(   __lowerCAmelCase							)				->   bool:




						'''simple docstring'''



						assert isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							) and (
						    number >= 0
						), "'number' must been an int and positive"

						lowerCamelCase__   =True

						# 0 and 1 are none primes.
						if number <= 1:
												lowerCamelCase__   =False

						for divisor in range(2  ,       int(round(sqrt(__lowerCAmelCase							)							)							) + 1							):
												# if 'number' divisible by 'divisor' then sets 'status'
												# of false and break up the loop.
												if number % divisor == 0:
																		lowerCamelCase__   =False
																		break

    # precondition
						assert isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							), "'status' must been from type bool"

						return status





def      lowerCamelCase_					(   __lowerCAmelCase							)				->   List[str]:




						'''simple docstring'''



						assert isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							) and (n > 2), "'N' must been an int and > 2"

						# beginList: contains all natural numbers from 2 up to N
						lowerCamelCase__   =list(range(2  ,       n + 1							)							)

						lowerCamelCase__   =[]  # this list will be returns.

						# actual sieve of erathostenes
						for i in range(len(__lowerCAmelCase							)							):
												for j in range(i + 1  ,       len(__lowerCAmelCase							)							):
																		if (begin_list[i] != 0) and (begin_list[j] % begin_list[i] == 0):
																								lowerCamelCase__   =0

    # filters actual prime numbers.
						lowerCamelCase__   =[x for x in begin_list if x != 0]

						# precondition
						assert isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							), "'ans' must been from type list"

						return ans





def      lowerCamelCase_					(   __lowerCAmelCase							)				->   Optional[int]:




						'''simple docstring'''



						assert isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							) and (n > 2), "'N' must been an int and > 2"

						lowerCamelCase__   =[]

						# iterates over all numbers between 2 up to N+1
						# if a number is prime then appends to list 'ans'
						for number in range(2  ,       n + 1							):
												if is_prime(__lowerCAmelCase							):
																		ans.append(__lowerCAmelCase							)

    # precondition
						assert isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							), "'ans' must been from type list"

						return ans





def      lowerCamelCase_					(   __lowerCAmelCase							)				->   Union[str, Any]:




						'''simple docstring'''



						assert isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							) and number >= 0, "'number' must been an int and >= 0"

						lowerCamelCase__   =[]  # this list will be returns of the function.

						# potential prime number factors.

						lowerCamelCase__   =2

						lowerCamelCase__   =number

						if number == 0 or number == 1:
												ans.append(__lowerCAmelCase							)

						# if 'number' not prime then builds the prime factorization of 'number'
						elif not is_prime(__lowerCAmelCase							):
												while quotient != 1:
																		if is_prime(__lowerCAmelCase							) and (quotient % factor == 0):
																								ans.append(__lowerCAmelCase							)
																								quotient /= factor
																		else:
																								factor += 1

						else:
												ans.append(__lowerCAmelCase							)

						# precondition
						assert isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							), "'ans' must been from type list"

						return ans





def      lowerCamelCase_					(   __lowerCAmelCase							)				->   Optional[int]:




						'''simple docstring'''



						assert isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							) and (
						    number >= 0
						), "'number' bust been an int and >= 0"

						lowerCamelCase__   =0

						# prime factorization of 'number'
						lowerCamelCase__   =prime_factorization(__lowerCAmelCase							)

						lowerCamelCase__   =max(__lowerCAmelCase							)

						# precondition
						assert isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							), "'ans' must been from type int"

						return ans





def      lowerCamelCase_					(   __lowerCAmelCase							)				->   List[Any]:




						'''simple docstring'''



						assert isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							) and (
						    number >= 0
						), "'number' bust been an int and >= 0"

						lowerCamelCase__   =0

						# prime factorization of 'number'
						lowerCamelCase__   =prime_factorization(__lowerCAmelCase							)

						lowerCamelCase__   =min(__lowerCAmelCase							)

						# precondition
						assert isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							), "'ans' must been from type int"

						return ans





def      lowerCamelCase_					(   __lowerCAmelCase							)				->   Union[str, Any]:




						'''simple docstring'''



						assert isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							), "'number' must been an int"
						assert isinstance(number % 2 == 0  ,       __lowerCAmelCase							), "compare bust been from type bool"

						return number % 2 == 0





def      lowerCamelCase_					(   __lowerCAmelCase							)				->   List[Any]:




						'''simple docstring'''



						assert isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							), "'number' must been an int"
						assert isinstance(number % 2 != 0  ,       __lowerCAmelCase							), "compare bust been from type bool"

						return number % 2 != 0





def      lowerCamelCase_					(   __lowerCAmelCase							)				->   int:




						'''simple docstring'''



						assert (
						    isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							) and (number > 2) and is_even(__lowerCAmelCase							)
						), "'number' must been an int, even and > 2"

						lowerCamelCase__   =[]  # this list will returned

						# creates a list of prime numbers between 2 up to 'number'
						lowerCamelCase__   =get_prime_numbers(__lowerCAmelCase							)
						lowerCamelCase__   =len(__lowerCAmelCase							)

						# run variable for while-loops.
						lowerCamelCase__   =0
						lowerCamelCase__   =None

						# exit variable. for break up the loops
						lowerCamelCase__   =True

						while i < len_pn and loop:
												lowerCamelCase__   =i + 1

												while j < len_pn and loop:
																		if prime_numbers[i] + prime_numbers[j] == number:
																								lowerCamelCase__   =False
																								ans.append(prime_numbers[i]							)
																								ans.append(prime_numbers[j]							)

																		j += 1

												i += 1

						# precondition
						assert (
						    isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							)
						    and (len(__lowerCAmelCase							) == 2)
						    and (ans[0] + ans[1] == number)
						    and is_prime(ans[0]							)
						    and is_prime(ans[1]							)
						), "'ans' must contains two primes. And sum of elements must been eq 'number'"

						return ans





def      lowerCamelCase_					(   __lowerCAmelCase  ,       __lowerCAmelCase							)				->   Tuple:




						'''simple docstring'''



						assert (
						    isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							)
						    and isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							)
						    and (numbera >= 0)
						    and (numbera >= 0)
						), "'number1' and 'number2' must been positive integer."

						lowerCamelCase__   =0

						while numbera != 0:
												lowerCamelCase__   =numbera % numbera
												lowerCamelCase__   =numbera
												lowerCamelCase__   =rest

						# precondition
						assert isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							) and (
						    numbera >= 0
						), "'number' must been from type int and positive"

						return numbera





def      lowerCamelCase_					(   __lowerCAmelCase  ,       __lowerCAmelCase							)				->   Union[str, Any]:




						'''simple docstring'''



						assert (
						    isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							)
						    and isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							)
						    and (numbera >= 1)
						    and (numbera >= 1)
						), "'number1' and 'number2' must been positive integer."

						lowerCamelCase__   =1  # actual answer that will be return.

						# for kgV (x,1)
						if numbera > 1 and numbera > 1:
												# builds the prime factorization of 'number1' and 'number2'
												lowerCamelCase__   =prime_factorization(__lowerCAmelCase							)
												lowerCamelCase__   =prime_factorization(__lowerCAmelCase							)

						elif numbera == 1 or numbera == 1:
												lowerCamelCase__   =[]
												lowerCamelCase__   =[]
												lowerCamelCase__   =max(__lowerCAmelCase  ,       __lowerCAmelCase							)

						lowerCamelCase__   =0
						lowerCamelCase__   =0

						lowerCamelCase__   =[]  # captured numbers int both 'primeFac1' and 'primeFac2'

						# iterates through primeFac1
						for n in prime_fac_a:
												if n not in done:
																		if n in prime_fac_a:
																								lowerCamelCase__   =prime_fac_a.count(__lowerCAmelCase							)
																								lowerCamelCase__   =prime_fac_a.count(__lowerCAmelCase							)

																								for _ in range(max(__lowerCAmelCase  ,       __lowerCAmelCase							)							):
																														ans *= n

																		else:
																								lowerCamelCase__   =prime_fac_a.count(__lowerCAmelCase							)

																								for _ in range(__lowerCAmelCase							):
																														ans *= n

																		done.append(__lowerCAmelCase							)

    # iterates through primeFac2
						for n in prime_fac_a:
												if n not in done:
																		lowerCamelCase__   =prime_fac_a.count(__lowerCAmelCase							)

																		for _ in range(__lowerCAmelCase							):
																								ans *= n

																		done.append(__lowerCAmelCase							)

    # precondition
						assert isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							) and (
						    ans >= 0
						), "'ans' must been from type int and positive"

						return ans





def      lowerCamelCase_					(   __lowerCAmelCase							)				->   Optional[Any]:




						'''simple docstring'''



						assert isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							) and (n >= 0), "'number' must been a positive int"

						lowerCamelCase__   =0
						lowerCamelCase__   =2  # this variable holds the answer

						while index < n:
												index += 1

												ans += 1  # counts to the next number

												# if ans not prime then
												# runs to the next prime number.
												while not is_prime(__lowerCAmelCase							):
																		ans += 1

    # precondition
						assert isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							) and is_prime(
						    __lowerCAmelCase							), "'ans' must been a prime number and from type int"

						return ans





def      lowerCamelCase_					(   __lowerCAmelCase  ,       __lowerCAmelCase							)				->   int:




						'''simple docstring'''



						assert (
						    is_prime(__lowerCAmelCase							) and is_prime(__lowerCAmelCase							) and (p_number_a < p_number_a)
						), "The arguments must been prime numbers and 'pNumber1' < 'pNumber2'"

						lowerCamelCase__   =p_number_a + 1  # jump to the next number

						lowerCamelCase__   =[]  # this list will be returns.

						# if number is not prime then
						# fetch the next prime number.
						while not is_prime(__lowerCAmelCase							):
												number += 1

						while number < p_number_a:
												ans.append(__lowerCAmelCase							)

												number += 1

												# fetch the next prime number.
												while not is_prime(__lowerCAmelCase							):
																		number += 1

    # precondition
						assert (
						    isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							)
						    and ans[0] != p_number_a
						    and ans[len(__lowerCAmelCase							) - 1] != p_number_a
						), "'ans' must been a list without the arguments"

						# 'ans' contains not 'pNumber1' and 'pNumber2' !
						return ans





def      lowerCamelCase_					(   __lowerCAmelCase							)				->   int:




						'''simple docstring'''



						assert isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							) and (n >= 1), "'n' must been int and >= 1"

						lowerCamelCase__   =[]  # will be returned.

						for divisor in range(1  ,       n + 1							):
												if n % divisor == 0:
																		ans.append(__lowerCAmelCase							)

    # precondition
						assert ans[0] == 1 and ans[len(__lowerCAmelCase							) - 1] == n, "Error in function getDivisiors(...)"

						return ans





def      lowerCamelCase_					(   __lowerCAmelCase							)				->   Union[str, Any]:




						'''simple docstring'''



						assert isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							) and (
						    number > 1
						), "'number' must been an int and >= 1"

						lowerCamelCase__   =get_divisors(__lowerCAmelCase							)

						# precondition
						assert (
						    isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							)
						    and (divisors[0] == 1)
						    and (divisors[len(__lowerCAmelCase							) - 1] == number)
						), "Error in help-function getDivisiors(...)"

						# summed all divisors up to 'number' (exclusive), hence [:-1]
						return sum(divisors[:-1]							) == number





def      lowerCamelCase_					(   __lowerCAmelCase  ,       __lowerCAmelCase							)				->   Dict:




						'''simple docstring'''



						assert (
						    isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							)
						    and isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							)
						    and (denominator != 0)
						), "The arguments must been from type int and 'denominator' != 0"

						# build the greatest common divisor of numerator and denominator.
						lowerCamelCase__   =gcd(abs(__lowerCAmelCase							)  ,       abs(__lowerCAmelCase							)							)

						# precondition
						assert (
						    isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							)
						    and (numerator % gcd_of_fraction == 0)
						    and (denominator % gcd_of_fraction == 0)
						), "Error in function gcd(...,...)"

						return (numerator // gcd_of_fraction, denominator // gcd_of_fraction)





def      lowerCamelCase_					(   __lowerCAmelCase							)				->   List[Any]:




						'''simple docstring'''



						assert isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							) and (n >= 0), "'n' must been a int and >= 0"

						lowerCamelCase__   =1  # this will be return.

						for factor in range(1  ,       n + 1							):
												ans *= factor

						return ans





def      lowerCamelCase_					(   __lowerCAmelCase							)				->   Tuple:




						'''simple docstring'''



						assert isinstance(__lowerCAmelCase  ,       __lowerCAmelCase							) and (n >= 0), "'n' must been an int and >= 0"

						lowerCamelCase__   =0
						lowerCamelCase__   =1
						lowerCamelCase__   =1  # this will be return

						for _ in range(n - 1							):
												lowerCamelCase__   =ans
												ans += fiba
												lowerCamelCase__   =tmp

						return ans


